Properties Of Operators Research Articles

Theoretical Investigation on the Conservation Principles of an Extended Davey–Stewartson System with Riesz Space Fractional Derivatives of Different Orders

In this article, a generalized form of the Davey–Stewartson system, consisting of three nonlinear coupled partial differential equations, will be studied. The system considers the presence of fractional spatial partial derivatives of the Riesz type, and extensions of the classical mass, energy, and momentum operators will be proposed in the fractional-case scenario. In this work, we will prove rigorously that these functionals are conserved throughout time using some functional properties of the Riesz fractional operators. This study is intended to serve as a stepping stone for further exploration of the generalized Davey–Stewartson system and its wide-ranging applications.

Solutions of linear and non-linear partial differential equations by means of tensor product theory of Banach space

In this article, we introduce an analytical method for solving both non-separable linear and non-linear partial differential equations, for which separation of variables method does not work. This method is based on the theory of tensor product in Banach spaces coupled with some properties of atoms operators. We provide some illustrative examples.
 For more information see https://ejde.math.txstate.edu/Volumes/2024/28/abstr.html

Injectable plant-derived polysaccharide hydrogels with intrinsic antioxidant bioactivity accelerate wound healing by promoting epithelialization and angiogenesis

Skin wound healing is a complex and dynamic process involving hemostasis, inflammatory response, cell proliferation and migration, and angiogenesis. Currently used wound dressings remain unsatisfactory in the clinic due to the lack of adjustable mechanical property for injection operation and bioactivity for accelerating wound healing. In this work, an “all-sugar” hydrogel dressing is developed based on dynamic borate bonding network between the hydroxyl groups of okra polysaccharide (OP) and xyloglucan (XG). Benefiting from the reversible crosslinking network, the resulting composite XG/OP hydrogels exhibited good shear-thinning and fast self-healing properties, which is suitable to be injected at wound beds and filled into irregular injured site. Besides, the proposed XG/OP hydrogels showed efficient antioxidant capacity by scavenging DPPH activity of 73.9 %. In vivo experiments demonstrated that XG/OP hydrogels performed hemostasis and accelerated wound healing with reduced inflammation, enhanced collagen deposition and angiogenesis. This plant-derived dynamic hydrogel offers a facile and effective approach for wound management and has great potential for clinical translation in feature.

Effect of silver chloride deposition technique on modified bentonite operating properties for radioactive iodide-ions localization in geological disposal facility for radioactive waste

This work discusses the influence of the technique of silver chloride deposition (via one-stage or two-stage method) onto performance attributes of modified bentonite for iodine ions localization in radioactive waste repositories. It has been established that bentonite modified via both techniques in wide range of AgCl content has high sorption properties (more than 95%) towards iodide anions, the AgCl deposition method has little-to-no effect on sorption properties. Investigation by scanning electron microscopy and energy dispersive X-ray spectroscopy methods and studying of silver leaching in various media showed that two-stage technique yield more uniform AgCl distribution and significantly higher resistance of AgCl to leaching than one-stage technique. Therefore, as a component of bentonite engineered safety barrier in geological disposal facility for radioactive waste use of bentonite modified with two-stage technique is preferable.

Operating Properties of Disease Activity Indices in Pediatric Inflammatory Bowel Disease: A Systematic Review.

Accurate, reliable, and responsive disease activity indices are important to streamline drug approval and treatment modalities for pediatric inflammatory bowel disease (pIBD). We aimed to identify all scoring indices used in pIBD randomized controlled trials (RCTs) and to evaluate their operating properties. MEDLINE, EMBASE, and CENTRAL were searched on December 6, 2022, to identify studies evaluating clinical, endoscopic, imaging, or patient-reported outcome measures (PROMs) in pIBD including Crohn's disease (CD) and ulcerative colitis (UC). Validity, reliability, responsiveness, and feasibility were summarized. Seventy RCTs evaluating pIBD indices were identified. Forty-one studies reported on the operating properties of 14 eligible indices (n = 9 CD, n = 5 UC). The Pediatric Crohn's Disease Activity Index (PCDAI) varied widely in terms of validity and reliability and was less feasible overall. In contrast, the Mucosal Inflammation Noninvasive Index, which includes fecal calprotectin, had better operating properties than the PCDAI. The Simplified Endoscopic Mucosal Assessment of Crohn's Disease appears more feasible and had similar operating properties than the longer Simple Endoscopic Score for Crohn's Disease. The Pediatric Ulcerative Colitis Activity Index was feasible, valid, and reliable, but responsiveness needs to be evaluated further. The Endoscopic Mayo score and the Ulcerative Colitis Endoscopic Index of Severity were reliable, but validity and responsiveness need to be evaluated further. Imaging and PROMs/quality of life indices need further evaluation. The operating properties of pIBD clinical trial end points varied widely. These results highlight the need for further validation and development of novel indices.

Operating Properties of Deep Hole Boring Tools with Modified Design.

This paper presents the results of research work on the revised design of a deep hole boring tool. The study was divided into three stages: theoretical, experimental and operational. In the theoretical part, a 3D model of the actual boring bar was created, which was subjected to strength tests using the Finite Element Method (FEM), and then prototypes of new deep hole boring tools were made with structural modifications to the shank part of the tool. For the polymer concrete core of a shank, there was a 14.59% lower displacement, and for the rubber-doped polymer concrete (SBR-styrene butadiene rubber) core of a shank there was a 4.84% lower displacement in comparison to the original boring bar. In the experimental part of the study, the original boring bar and the prototypes were subjected to experimental modal analysis and static analysis tests to compare dynamic and static properties. In the operational part of the study, boring tests were carried out for various workpiece materials, during which the basic parameters of the surface geometric structure (SGS), such as roughness Ra and Rz, were studied. Despite the promising preliminary results of the theoretical and experimental studies, using the described modifications to the boring bar is not recommended.

THE EFFECT OF MODIFYING ADDITIVES ON THE PER-FORMANCE PROPERTIES OF SLAG-ALKALI BINDERS AND CONCRETES

The results of the analysis and generalization of research on the problem of using man-made industrial waste for the production of composite slag-alkali binders and concretes based on them are presented. The disposal of industrial man-made waste and their use in the production of composite binders are issues of global importance. Therefore, much attention is paid to this issue in the industrialized countries of the world.The influence of waste products from the enrichment of polymetallic ores – carbonate-barium tailings on the performance properties (strength, water absorption, density, etc.) of composite slag-alkali binders and concretes based on them has been studied. Various methods of introducing modifying additives into the composition of composite slag-alkali binders and concretes based on them have been investigated. The optimal amount of introduction of modifying additives into the composition of composite slag–alkali binders - waste of carbonate-barium tailings and an efficiency indicator for the effect on the activity of concrete, depending on the type of alkaline component and hardening conditions, has been established. The methods of conducting experimental work to determine the operational properties of slag-alkali binders and concretes based on them are presented.

Research of physical and mechanical properties of knitted fabrics of plush interlooping with reduced material consumption

The article discusses methods for manufacturing lined weave plush knitwear and ways to reduce the material intensity of knitted fabrics. The purpose of the study is to reduce the material consumption of products while maintaining operational, hygienic and aesthetic properties in the production of knitted fabrics. As a result of experimental work, the technological capabilities of the KН-323D flat knitting machine were studied and new types of lined weave plush knitwear were developed. Comparing the bulk density of plush knitwear with elongated lining broaches with the main base weave, it is clear that the presence of elongated lining broaches in the knitwear structure increases the thickness of the knitwear and reduces the bulk density. The breathability of a lined weave plush jersey is significantly less than that of a basic weave. It has been established that air permeability coefficients can be reduced to 33.1% (46.3-66.9 cm3/cm2∙sec) by adjusting the length of the threads of the lining broaches of plush knitted fabric of lined weave. Regression equations are derived to determine the dependence of bulk density, breaking load (along the length and width) and the air permeability index of knitted fabric on technological factors. Mathematical models were built in Excel using regression equations. The research results make it possible to improve the quality and consumer properties of knitted products, and to select advanced production technologies for their production.

Curved Representation Space of Vision Transformers

Neural networks with self-attention (a.k.a. Transformers) like ViT and Swin have emerged as a better alternative to traditional convolutional neural networks (CNNs). However, our understanding of how the new architecture works is still limited. In this paper, we focus on the phenomenon that Transformers show higher robustness against corruptions than CNNs, while not being overconfident. This is contrary to the intuition that robustness increases with confidence. We resolve this contradiction by empirically investigating how the output of the penultimate layer moves in the representation space as the input data moves linearly within a small area. In particular, we show the following. (1) While CNNs exhibit fairly linear relationship between the input and output movements, Transformers show nonlinear relationship for some data. For those data, the output of Transformers moves in a curved trajectory as the input moves linearly. (2) When a data is located in a curved region, it is hard to move it out of the decision region since the output moves along a curved trajectory instead of a straight line to the decision boundary, resulting in high robustness of Transformers. (3) If a data is slightly modified to jump out of the curved region, the movements afterwards become linear and the output goes to the decision boundary directly. In other words, there does exist a decision boundary near the data, which is hard to find only because of the curved representation space. This explains the underconfident prediction of Transformers. Also, we examine mathematical properties of the attention operation that induce nonlinear response to linear perturbation. Finally, we share our additional findings, regarding what contributes to the curved representation space of Transformers, and how the curvedness evolves during training.

К ФОРМИРОВАНИЮ МОДЕЛИ ЭФФЕКТИВНОСТИ ФУНКЦИО-НИРОВАНИЯ СЕЛЬСКОХОЗЯЙСТВЕННОГО ТРАКТОРА В СОСТАВЕ МТА ПРИ РАБОТЕ НА АЛЬТЕРНАТИВНЫХ ВИДАХ ТОПЛИВА

The development of the Russian agro-industrial complex provides for increasing the efficiency of the functioning of energy machines while simultaneously reducing energy and labor costs. A promising direction for saving diesel fuel and improving the environmental performance of mobile agricultural machinery is the use of alternative motor fuel. In modern agricultural machines, this will be reflected, first of all, in the parameters of the working process and its flow in the engine, as well as in the indicators of power, fuel efficiency and environmental friendliness of its operation, which, to one degree or another, will lead to changes in the indicators characterizing the operational properties of a machine-tractor unit. The functioning of the engine as a model can be presented in the form of a multi-level and multidimensional system in which the control flows will be indicators of the work process (effective power, indicator pressure, the amount of fresh charge entering the cylinders, etc.), fuel supply parameters (cyclic supply, fuel injection pressure, fuel temperature, etc.), which depend on the physicochemical and operational properties of the alternative fuel. The proposed model provides for the determination of the rational composition of the fuel, technical solutions for adapting the fuel system for the effective supply of alternative fuel to the engine cylinders with the required indicators, as well as determining their influence on the parameters of the operational efficiency of the tractor as part of machine-tractor unit. The obtained numerical values of the parameters of the diesel operating process and fuel supply, depending on the physical-chemical properties of the alternative fuel used, make it possible to determine the performance indicators of the tractor as part of machine-tractor unit and evaluate the efficiency of its operation on various alternative types of fuels, in comparison with using petroleum diesel fuel.

On the Eigenvalues of a Soluble Model of Coupled Molecular and Nuclear Hamiltonians

In this paper, a special type of soluble model corresponding to a coupled molecular and nuclear Hamiltonians H, the so-called generalized Friedrichs model, is considered. We aim to determine and provide the most important properties of the well-known Faddeev operator corresponding to H accommodating a number of discrete eigenvalues. Furthermore, we provide a formula for counting the multiplicity of discrete eigenvalues of H.

Адаптация системы ЧПУ станка к условиям комбинированной обработки

Introduction. Increasing the efficiency of processing technologies for products made from modern high-strength, difficult-to-process materials with increased physical, mechanical and operational properties consists not only in improving the technology itself, the tools for its implementation, but also in modernizing technological equipment taking into account new achievements in the field of mechanical engineering. Modern computer numerical control (CNC) equipment is now quite advanced in terms of controlling basic cutting movements. Adaptive monitoring and control systems, as a rule, additionally installed on processing equipment, make it possible to further improve the quality of processing parameters. With the development of new hybrid and combined technologies that combine several types of influence on the product being processed, the issue of synchronizing the automatic control of the movements of parts of technological equipment with the control and management of accompanying processes of combined technologies has become acute. One example of such technologies is electrochemical diamond grinding with periodic dressing of the working surface of a diamond wheel using reverse polarity current. The polarity of the current and the duration of its pulses are controlled by special programmable devices. Current switching units are connected to it. It serves to supply alternating currents of direct and reverse polarity to the electrical circuit and is made on the basis of key elements. Installing such programmable devices on CNC machines leads to its’ equipping with an additional autonomous automatic control system. At the same time, it is difficult to coordinate the operation of the machine’s CNC system, which controls the movements of its working parts, and the programmable device used to control the polarity and duration of current pulses during combined processing. The purpose of the work is to synchronize the CNC system of the machine with the control system for the process of periodically changing the polarity of the current. The study was carried out on an experimental stand. Methods. The research methodology involved conducting an experiment consisting of synchronizing the operation of the machine’s CNC system with the operation of the control system for the process of periodically changing the polarity of the current. To evaluate the results, the time of movement of the diamond wheel as a result of the working stroke was compared with the duration of current pulses of different polarities specified in the control program of the developed software. Results and discussions. As a result of the research, it is established that the developed software and hardware complex makes it possible to synchronize in the CNC system of the machine tool the control of the movements of the working parts with automatic control of the periodic change of current polarity during electrochemical diamond grinding, which can significantly expand the technical capabilities of CNC machines.

New Results on (r,k,μ)-Riemann–Liouville Fractional Operators in Complex Domain with Applications

This paper introduces fractional operators in the complex domain as generalizations for the Srivastava–Owa operators. Some properties for the above operators are also provided. We discuss the convexity and starlikeness of the generalized Libera integral operator. A condition for the convexity and starlikeness of the solutions of fractional differential equations is provided. Finally, a fractional differential equation is converted into an ordinary differential equation by wave transformation; illustrative examples are provided to clarify the solution within the complex domain.

Chimera states in a lattice of superdiffusively coupled neurons

Chimera states are a truly remarkable dynamical phenomenon that occur in systems of coupled oscillators. In this regime, regions of synchronized and unsynchronized elements are formed in the system. For many applied problems, especially in neuroscience, these states offer a rich potential for research. However, the plethora of models and the lack of a ”single simple principle” that leads to the development of chimeras makes it very difficult to understand their nature. In this work, we propose a three-component reaction-superdiffusion system based on a unified mechanism founded on the properties of the fractional Laplace operator and the nonlinear Hindmarsh-Rose model functions. In the proposed system, the non-local type of interaction forming the coupling between the elements depends significantly on the fractional Laplace operator exponents of the corresponding components. It is shown that in the framework of the superdiffusion type of interaction, chimera states are realized in the system. At the same time, many qualitative (shape, visual degree of inhomogeneity and area size) and quantitative characteristics of chimeras (synchronization factor, strength of incoherence, local order parameter, number of elements with a potential value exceeding a given one) depend significantly on the exponents of the fractional Laplace operator. In addition to classical chimeras and target-waves chimeras, the results of numerical simulations show the presence of mutually sustaining reaction processes of different scales in the system.

Characterization of Second-Order Mixing Effects in Reconstructed Cross-Spectra of Random Neural Fields

Functional connectivity in electroencephalography (EEG) and magnetoencephalography (MEG) data is commonly assessed by using measures that are insensitive to instantaneously interacting sources and as such would not give rise to false positive interactions caused by instantaneous mixing of true source signals (first-order mixing). Recent studies, however, have drawn attention to the fact that such measures are still susceptible to instantaneous mixing from lagged sources (i.e. second-order mixing) and that this can lead to a large number of false positive interactions. In this study we relate first- and second-order mixing effects on the cross-spectra of reconstructed source activity to the properties of the resolution operators that are used for the reconstruction. We derive two identities that relate first- and second-order mixing effects to the transformation properties of measurement and source configurations and exploit them to establish several basic properties of signal mixing. First, we provide a characterization of the configurations that are maximally and minimally sensitive to second-order mixing. It turns out that second-order mixing effects are maximal when the measurement locations are far apart and the sources coincide with the measurement locations. Second, we provide a description of second-order mixing effects in the vicinity of the measurement locations in terms of the local geometry of the point-spread functions of the resolution operator. Third, we derive a version of Lagrange’s identity for cross-talk functions that establishes the existence of a trade-off between the magnitude of first- and second-order mixing effects. It also shows that, whereas the magnitude of first-order mixing is determined by the inner product of cross-talk functions, the magnitude of second-order mixing is determined by a generalized cross-product of cross-talk functions (the wedge product) which leads to an intuitive geometric understanding of the trade-off. All results are derived within the general framework of random neural fields on cortical manifolds.

Browder operators on quaternionic Hilbert space

In this paper, we study the notions of Browder operator and Browder S-spectrum of bounded right linear operator defined over the right quaternionic Hilbert space. Some properties of Browder operator and stability of the ascent, descent and Browder S-spectrum have been investigated in the right quaternionic setting. We also characterize the property of invariant Browder operators and study the spectral mapping theorem of Browder S-spectrum for self-adjoint operators in quaternionic setting. This investigation concludes by exploring the Browder S-spectrum of the sum of two bounded linear operators.

Convolution Kernel Function and Its Invariance Properties of Bone Fractal Operators

This article studies the error function and its invariance properties in the convolutional kernel function of bone fractal operators. Specifically, the following contents are included: (1) demonstrating the correlation between the convolution kernel function and error function of bone fractal operators; (2) focusing on the main part of bone fractal operators: p+α2-type differential operator, discussing the convolutional kernel function image; (3) exploring the fractional-order correlation between the error function and other special functions from the perspective of fractal operators.

Justification of the criterion for technical synthesis of radar complexes for covert airspace control

For covert control of airspace, it is required to use complexes that meet the requirements for the effectiveness of obtaining data on objects in the field of observations and protection against the detection of their own unmasking signs by means of radio technical monitoring with predetermined indicators of resource intensity and efficiency of performing target functions. Joint fulfillment of these requirements when the parameters of the equipment structures and information processes are found under the established conditions for monitoring the properties of resource intensity and efficiency is achieved. Due to the complexity and inconsistency of the dependent bridges of the performance indicators and the stealth of application from the technical characteristics of the radar systems, it is necessary to substantiate the synthesis criterion for finding compromise technical solutions for the design of the equipment and the choice of a strategy for obtaining data on air facilities. The goal of the article is determination of design characteristics, frequency-time and energy signals parameters of radar complexes for analysis of air situation at concealment of unmasking features from means of radio technical monitoring. The criterion of technical synthesis of radar complexes for covert control of air-space is justified, which allows obtaining information about air objects at pre-established risks of threats to identify their own unmasking signs by means of radio technical monitoring. Regularities have been investigated and the conditions for achieving the target functions of complexes with resonant, band and frequency-independent signal transmission-reception devices have been determined. The principles of construction have been substantiated and the rules for finding rational parameters of the design of radar complexes and information processes used in airspace control in order to achieve the required operational properties of obtaining information about air objects while maintaining secrecy from radio monitoring facilities have been determined.

Cubic spline quasi-interpolation operator to numerically solve integro-differential equations with weakly singular kernels

In this paper we use cubic spline quasi-interpolant operator to numerically address a class of linear integro-differential equations with weakly singular kernel. As stated in Pedas and Tamme (2006), the exact solution of this equation lacks the desired level of smoothness and belongs to a particular function space. Then, in the first part of this paper, we analyze the approximation properties of the cubic spline quasi-interpolant operator in this particular space. Subsequently, we use these results in the analysis of the quasi-collocation method used to solve an integro-differential equation with weakly singular kernel. Also, some numerical tests are provided to confirm the theoretical results.

Approximation by frame-like multiwavelets

Approximation properties of multivariate quasi-projection operators [Formula: see text] generated by vectors of compactly supported functions [Formula: see text], [Formula: see text] are studied. Error estimates in [Formula: see text]-norm are obtained for a wide class of such operators. For refinable function vectors [Formula: see text], [Formula: see text] quasi-projection operators are related to dual multiwavelet systems. Although the general scheme for the construction of dual multiwavelet frames in the multivariate case is known, its realization in practice is a difficult task because of the necessity of providing some additional properties. The notion of frame-like multiwavelets is a relaxed version of multiwavelet frames. Frame-like multiwavelets retain frame-type decompositions waiving the usual frame condition. This simplifies the problem of frame-like multiwavelets construction. Approximation properties of frame-like multiwavelets are established. Algorithms for constructing frame-like multiwavelets with the desired approximation order are suggested.